Shallow circuits and concise formulae for multiple addition and multiplication

A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards.The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {, , }-circuits of depth 3.48 log₂ n and {, , }-circuits of depth 4.95 log₂ n for the carry-save addition ofn numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log₂ n and 5.95 log₂ n.We also get {, , }-formulae of sizeO (n ^3.13) and {, }-formulae of sizeO (n ^4.57) for all the output bits of a carry-save addition ofn numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.     

Optimal algorithms for symmetry detection in two and three dimensions

Exact algorithms for detecting all rotational and involutional symmetries in point sets, polygons and polyhedra are described. The time complexities of the algorithms are shown to be (n) for polygons and (n logn) for two- and three-dimensional point sets. (n logn) time is also required for general polyhedra, but for polyhedra with connected, planar surface graphs (n) time can be achieved. All algorithms are optimal in time complexity, within constants.     

Constructing strongly convex hulls using exact or rounded arithmetic

One useful generalization of the convex hull of a setS ofn points is the -strongly convex -hull. It is defined to be a convex polygon with vertices taken fromS such that no point inS lies farther than outside and such that even if the vertices of are perturbed by as much as , remains convex. It was an open question as to whether an -strongly convexO()-hull existed for all positive . We give here anO(n logn) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an -strongly convexO( + )-hull inO(n logn) time using rounded arithmetic with rounding unit . This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent ofn.     

Three ∑ P 2 -complete problems in computational learning theory

The consistency problem associated with a concept classC is to determine, given two setsA andB of examples, whether there exists a conceptc inC such that eachx inA is a positive example ofc and eachy inB is a negative example ofc. We explore in this paper the following intuition: for a concept classC, if the membership problem of determining whether a given example is positive for a concept isNP-complete, then the corresponding consistency problem is likely to be _P ² -complete. To support this intuition, we prove that the following three consistency problems for concept classes of patterns, graphs and generalized Boolean formulas, whose membership problems are known to beNP-complete, are _P ² -complete: (a) given two setsA andB of strings, determine whether there exists a patternp such that every string inA is in the languageL(p) and every string inB is not in the languageL(p); (b) given two setsA andB of graphs, determine whether there exists a graphG such that every graph inA is isomorphic to a subgraph ofG and every graph inB is not isomorphic to any subgraph ofG; and (c) given two setsA andB of Boolean formulas, determine whether there exists a 3-CNF Boolean formula such that for every A, is satisfiable and for every B, is not satisfiable. These results suggest that consistendy problems in machine learning are natural candidates for _P ² -complete problems if the corresponding membership problems are known to beNP-complete.In addition, we prove that the corresponding prediction problems for concept classes of polynomial-time nondeterministic Turing machines, nondeterministic Boolean circuits, generalized Boolean formulas, patterns and graphs are prediction-complete for the classR _NP of all concept classes whose membership problems are inNP.     

Undirecteds-t connectivity in polynomial time and sublinear space

Thes-t connectivity problem for undirected graphs is to decide whether two designated vertices,s andt, are in the same connected component. This paper presents the first known deterministic algorithms solving undirecteds-t connectivity using sublinear space and polynomial time. Our algorithms provide a nearly smooth time-space tradeoff between depth-first search and Savitch's algorithm. Forn vertex,m edge graphs, the simplest of our algorithms uses spaceO(s),n ^1/2log₂ nsnlog₂ n, and timeO(((m+n)n ² log² n)/s). We give a variant of this method that is faster at the higher end of the space spectrum. For example, with space (nlogn), its time bound isO((m+n)logn), close to the optimal time for the problem. Another generalization uses less space, but more time: spaceO(n ^1/logn), for 2log₂ n, and timen ^O(). For constant the time remains polynomial.     

Dynamic half-space range reporting and its applications

We consider the half-space range-reporting problem: Given a setS ofn points in ^d, preprocess it into a data structure, so that, given a query half-space , allk points ofS can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertions and deletions of points ofS. For a given parameterm,n m n ^d/2 and an arbitrarily small positive constant , we achieveO(m ¹⁺) space and preprocessing time, O((n/m ^d/2 logn+k) query time, and O(m¹⁺n) amortized update time (d 3). We present, among others, the following applications: an O(n¹⁺)-time algorithm for computing convex layers in ³, and an output sensitive algorithm for computing a level in an arrangements of planes in ³, whose time complexity is O((b+n) n, whereb is the size of the level.Work by the first author has been supported by National Science Foundation Grant CCR-91-06514. A preliminary version of this paper appeared in Agarwalet al. [2], which also contains the results of [20] on dynamic bichromatic closest pair and minimum spanning trees.     

The Use of Certain Equations of Mathematical Physics in Optimization of a Function on a Set. I

The results of application of potential theory to optimization are used to extend the use of (Helmholtz) diffusion and diffraction equations for optimization of their solutions (x, ) with respect to both x, and . If the aim function is modified such that the optimal point does not change, then the function (x, ) is convex in (x, for small . The possibility of using heat conductivity equation with a simple boundary layer for global optimization is investigated. A method is designed for making the solution U(x,t) of such equations to have a positive-definite matrix of second mixed derivatives with respect to x for any x in the optimization domain and any small t < 0 (the point is remote from the extremum) or a negative-definite matrix in x (the point is close to the extremum). For the functions (x, ) and U(x,t) having these properties, the gradient and the Newton–Kantorovich methods are used in the first and second stages of optimization, respectively.     

Minimization of norms and logarithmic norms by diagonal similarities

It is a commonly occurring problem to find good norms · or logarithmic norms (·) for a given matrix in the sense that they should be close to respectively the spectral radius (A) and the spectral abscissa (A). Examples may be the certification thatA is convergent, i.e. (A)A<1 or stable, i.e. (A)(A)<0. Often the ordinary norms do not suffice and one would like to try simple modifications of them such as using an ordinary norm for a diagonally transformed matrix. This paper treats this problem for some of the ordinary norms.

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Minimisierung von Normen und Logarithmischen Normen durch Diagonale Transformationen

Zusammenfassung Ein oft vorkommendes praktisches Problem ist die Konstruktion von guten Normen · und logarithmischen Normen (·) für eine gegebene MatrixA. Mit gut wird dann verstanden, daß A den Spektralradius (A)=max |₁| und (A) die Spektralabszisse (A)=max Re _i gut approximieren. Beispiele findet man für konvergente Matrizen wo (A)A<1 gewünscht ist, und für stabile Matrizen wo (A)(A)<0 zu zeigen ist. Wir untersuchen hier, wie weit man mit Diagonaltransformationen und dengewöhnlichsten Normen kommen kann.

     

Using geometry to solve the transportation problem in the plane

LetU andV be two sets of points in the plane, where ¦U¦=k,¦V¦=, andn=k+. These two sets of points induce a directed complete bipartite graph in which the points represent nodes and an edge is directed from each node inU to each node in K Each edge is given a cost equal to the distance between the corresponding nodes measured by some metricd on the plane. We consider thetransportation problem on such a graph. We present an 0(n^2,5 logn logN) algorithm, whereN is the magnitude of the largest supply or demand. The algorithm uses some fundamental results of computational geometry and scaling of supplies and demands. The algorithm is valid for the ₁ metric, the ₂ metric, and the metric. The running time for the ₁ and metrics can be improved to 0(n²(logn)³ logN).D. S. Atkinson was supported by the National Science Foundation under Grant CCR90-57481PYI. P. M. Vaidya was supported by the National Science Foundation under Grants CCR-9057481 and CCR-9007195.     

Simple adaptive stabilization of output feedback stabilizable distributed parameter systems

We show that the simple universal adaptive control lawu(t)=N(k(t))y(t)=|y(t)| ², withN(k)=(logk) cos((logk)) and 3+<1, stabilizes all detectable and stabilizable infinite dimensional systems of Pritchard-Salamon type which are externally stabilized by somescalar output feedback. The same controller is also shown to stabilize time varying systems satisfying the same type of output feedback stabilizability.     

A sublinear algorithm for approximate keyword searching

Given a relatively short query stringW of lengthP, a long subject stringA of lengthN, and a thresholdD, theapproximate keyword search problem is to find all substrings ofA that align withW with not more than D insertions, deletions, and mismatches. In typical applications, such as searching a DNA sequence database, the size of the databaseA is much larger than that of the queryW, e.g.,N is on the order of millions or billions andP is a hundred to a thousand. In this paper we present an algorithm that given a precomputedindex of the databaseA, finds rare matches in time that issublinear inN, i.e.,N ^c for somec<1. The sequenceA must be overa. finite alphabet . More precisely, our algorithm requires 0(DN ^pow() logN) expected-time where =D/P is the maximum number of differences as a percentage of query length, and pow() is an increasing and concave function that is 0 when =0. Thus the algorithm is superior to current O(DN) algorithms when is small enough to guarantee that pow() < 1. As seen in the paper, this is true for a wide range of , e.g., . up to 33% for DNA sequences (¦¦=4) and 56% for proteins sequences (¦¦=20). In preliminary practical experiments, the approach gives a 50-to 500-fold improvement over previous algorithms for prolems of interest in molecular biology.This work was supported in part by the National Institutes of Health under Grant R01 LM04960-01 and the Aspen Center for Physics.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

Drawings of graphs on surfaces with few crossings

We give drawings of a complete graphK _n withO(n ⁴ log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. We use these drawings ofK _n and give a polynomial-time algorithm for drawing any graph withn vertices andm edges withO(m ² log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genusg 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor ofO(log² g) from the lower bound (and hence from the optimal) for any graph withm 8n andn ²/m g m/64.The research of the third and the fourth authors was partially supported by Grant No. 2/1138/94 of the Slovak Academy of Sciences and by EC Cooperative action IC1000 Algorithms for Future Technologies (Project ALTEC). A preliminary version of this paper was presented at WG93 and published in Lecture Notes in Computer Science, Vol. 790, 1993, pp. 388–396.     

On communication-bounded synchronized alternating finite automata

We continue the study of communication-bounded synchronized alternating finite automata (SAFA), first considered by Hromkovi et al. We show that to accept a nonregular language, an SAFA needs to generate at least (log logn) communication symbols infinitely often; furthermore, a synchronized alternating finite automaton without nondeterminism (SUFA) needs to generate at least(log logn) communication symbols infinitely often for some constantk1. We also show that these bounds are tight.Next, we establish dense hierarchies of these machines on the function bounding the number of communication symbols. Finally, we give a characterization of NP in terms of communication-bounded multihead synchronized alternating finite automata, namely, NP = _k1 L(SAFA(k-heads,n ^k -com)). This result recasts the relationships between P, NP, and PSPACE in terms of multihead synchronized alternating finite automata.Research supported in part by NSF Grant CCR89-18409     

Nonrobustness Property of the Individual Ergodic Theorem

Main laws of probability theory, when applied to individual sequences, have a robustness property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length n grows as o(n). The law of iterated logarithm holds if the randomness deficiency grows as o(loglogn). We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon–McMillan–Breiman theorem.     

Parallel general prefix computations with geometric,algebraic, and other applications

We introduce a generic problem component that captures the most common, difficult kernel of many problems. This kernel involves general prefix computations (GPC). GPC's lower bound complexity of (n logn) time is established, and we give optimal solutions on the sequential model inO(n logn) time, on the CREW PRAM model inO(logn) time, on the BSR (broadcasting with selective reduction) model in constant time, and on mesh-connected computers inO(n) time, all withn processors, plus anO(log² n) time solution on the hypercube model. We show that GPC techniques can be applied to a wide variety of geometric (point set and tree) problems, including triangulation of point sets, two-set dominance counting, ECDF searching, finding two-and three-dimensional maximal points, the reconstruction of trees from their traversals, counting inversions in a permutation, and matching parentheses.work partially supported by NSF IRI/8709726work partially supported by NSERC.     

Robust algorithms for packet routing in a mesh

This paper considers the problem of permutation packet routing on a n×n mesh-connected array of processors. Each node in the array is assumed to be independently faulty with a probability bounded above by a valuep. This paper gives a routing algorithm which, ifp 0.29, will with very high probability route every packet that can be routed inO(n logn) steps with queue lengths that areO(log² n). Extensions to higher-dimensional meshes are given.     

Backtracking Problem in the Traversal of an Unknown Directed Graph by a Finite Robot

A covering path in a directed graph is a path passing through all vertices and arcs of the graph, with each arc being traversed only in the direction of its orientation. A covering path exists for any initial vertex only if the graph is strongly connected. The traversal of an unknown graph implies that the topology of the graph is not a priori known, and we learn it only in the course of traversing the graph. This is similar to the problem of traversing a maze by a robot in the case where the plan of the maze is not available. If the robot is a general-purpose computer without any limitations on the number of its states, then traversal algorithms with the estimate O(nm) are known, where n is the number of vertices and m is the number of arcs. If the number of states is finite, then this robot is a finite automaton. Such a robot is an analogue of the Turing machine, where the tape is replaced by a graph and the cells are assigned to the graph vertices and arcs. The selection of the arc that has not been traversed yet among those originating from the current vertex is determined by the order of the outgoing arcs, which is a priori specified for each vertex. The best known traversal algorithms for a finite robot are based on constructing the output directed spanning tree of the graph with the root at the initial vertex and traversing it with the aim to find all untraversed arcs. In doing so, we face the backtracking problem, which consists in searching for all vertices of the tree in the order inverse to their natural partial ordering, i.e., from the leaves to the root. Therefore, the upper estimate of the algorithms is different from the optimal estimate O(nm) by the number of steps required for the backtracking along the outgoing tree. The best known estimate O(nm + n ²loglogn) has been suggested by the author in the previous paper [1]. In this paper, a finite robot is suggested that performs a backtracking with the estimate O(n ²log*(n)). The function log* is defined as an integer solution of the inequality 1 log₂ ^log*(n) < 2, where log ^t = log º log º ... º log (the superposition º is applied t – 1 times) is the tth compositional degree of the logarithm. The estimate O(nm + n ²log*(n)) for the covering path length is valid for any strongly connected graph for a certain (unfortunately, not arbitrary) order of the outgoing arcs. Interestingly, such an order of the arcs can be marked by symbols of the finite robot traversing the graph. Hence, there exists a robot that traverses the graph twice: first traversal with the estimate O(nm + n ²loglogn) and the second traversal with the estimate O(nm + n ²log*(n)).     

The existence of solutions,stability, and linearization of volterra systems

LetB be a Banach space ofR _n valued continuous functions on [0, ) withfB. Consider the nonlinear Volterra integral equation (^*)x(t)+ _o ^t K(t,s,x(s))ds. We use the implicit function theorem to give sufficient conditions onB andK (t,s,x) for the existence of a unique solutionxB to (^*) for eachf B with f ^B sufficiently small. Moreover, there is a constantM>0 independent off with Mf^B.Part of this work was done while the author was visiting at Wright State University.